Theorems REF  and REF  imply that both Bonferroni and FDR type inference are asymptotically optimal only when \(\beta >(7-\sigma ^2)/8\) . This situation is similar to the case of the Gaussian means model studied in [1]}, implying that under moderate sparsity the evidence for discriminating \(H_0\)  from \(H_1^{(n)}\)  are not amongst sets of the form \(\lbrace p_i,\,:\,p_i < qk/n\rbrace \) , \(q\in (0,1)\) , \(k=1,\ldots ,n\) . Asymptotically, optimal discrimination is achieved by considering P-values in the much wider range \(\lbrace p_i,\,:\, p_i < n^{-(1-\delta )}\rbrace \)  for some \(\delta >0\) . This range is considered by HC and BJ, but not by FDR or Bonferroni.
